Entire solutions of certain type of difference equations

Journal of Inequalities and Applications, Feb 2014

In this paper, we shall study the conditions regarding the existence of transcendental entire solutions of certain type of difference equations. Our results are either supplements to some results obtained recently, or are relating to the conjecture raised in Yang and Laine (Proc. Jpn. Acad., Ser. A, Math. Sci. 86:10-14, 2010). Finally, two relevant conjectures are posed for further studies. MSC: 39B32, 34M05, 30D35.

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Entire solutions of certain type of difference equations

Journal of Inequalities and Applications Entire solutions of certain type of difference equations Nana Liu 0 Weiran Lü 0 Tiantian Shen 0 Chungchun Yang 1 0 Department of Mathematics, China University of Petroleum , Qingdao, 266580 , P.R. China 1 Department of Mathematics, Nanjing University , Nanjing, 210093 , P.R. China In this paper, we shall study the conditions regarding the existence of transcendental entire solutions of certain type of difference equations. Our results are either supplements to some results obtained recently, or are relating to the conjecture raised in Yang and Laine (Proc. Jpn. Acad., Ser. A, Math. Sci. 86:10-14, 2010). Finally, two relevant conjectures are posed for further studies. MSC: 39B32; 34M05; 30D35 - Consider a transcendental meromorphic function f (z) and let P(z, f ) = where al(z) (l = , , . . . , n) are small functions of f , and clj (l = , , . . . , n; j = , , . . . , k) are complex constants, and nlj (l = , , . . . , n; j = , , . . . , k) are natural numbers. nlj = nl j (j = , , . . . , k). Group together similar terms of P(z, f ), if necessary. In the following, we assume that no two terms of P(z, f ) are similar and that al(z) ≡  (l = , , . . . , n). Definition . We define the total degree d of P(z, f ) Yang and Laine [] considered the following difference equation and proved it. Theorem A A non-linear difference equation f (z) + q(z)f (z + ) = c sin bz, where q(z) is a non-constant polynomial and b, c ∈ C are nonzero constants, does not admit entire solutions of finite order. If q(z) = q is a nonzero constant, then the above equation possesses three distinct entire solutions of finite order, provided that b = nπ and q = (–)n+c/ for a nonzero integer n. Now, we shall substitute f (z + ) by f (z) in Theorem A and prove the following results. Theorem . Let n ≥  be an integer, q(z) be a polynomial, and p, p, α, α be nonzero constants such that α = α. If there exists some entire solution f of finite order to (.) below then q(z) is a constant, and one of the following relations holds: α (()) ff ((zz)) == ccee αnnzz,,aannddcc((eeαα/n/n––))qq==pp,,αα==nnαα,, where c, c are constants satisfying cn = p, cn = p. Corollary . Let n ≥  be an integer, q(z) ≡  be a polynomial, and p, α be nonzero constants. Then the non-linear difference equation By some further analysis, we can derive the following result. Theorem . A non-linear difference equation f (z) + q(z) f (z) = c sin bz, where q(z) is a non-constant polynomial and b, c ∈ C are nonzero constants, does not admit entire solutions of finite order. If q(z) = q is a nonzero constant, then (.) possesses solutions of the form f (z) = ce bi z + ce– bi z, c = –c = ci , provided that b = nπ , n is an odd number, q = , c or b = nπ ± π , q = –  c. Examples In the special case of a finite order entire solution is f (z) = i sin π z = eiπz – e–iπz. And f (z) +  f (z) = i sin π z has the entire solution f (z) = e π iz – e– π iz. Corollary . The non-linear difference equation f n + q f (z) = c sin bz, f (z) + q(z)f (z + ) = p(z) has no transcendental entire solutions of finite order. We shall modify the equation in Theorem B above and derive the following result. Theorem . Let l ≥ , n ≥  be integers, a(z) and b(z) be meromorphic functions such that T (r, a) = λT (r, f ) + S(r, f ), T (r, b) = λT (r, f ) + S(r, f ), where non-negative numbers λ, λ satisfy λ + λ < . Then f l(z) + a(z) nf (z) = b(z) 2 Preliminaries Lemma . ([]) Let f be a transcendental meromorphic solution of finite order ρ of a difference equation of the form H(z, f )P(z, f ) = Q(z, f ), where H(z, f ), P(z, f ), Q(z, f ) are difference polynomials in f such that the total degree of H(z, f ) in f and its shifts is n, and that the corresponding total degree of Q(z, f ) is ≤ n. If H(z, f ) contains just one term of maximal total degree, then for any ε > , possibly outside of an exceptional set of finite logarithmic measure. Remark . The following result is a Clunie type lemma [] for the difference-differential polynomials of a meromorphic function f . It can be proved by applying Lemma . with a similar reasoning as in [] and stated as follows. Let f (z) be a meromorphic function of finite order, and let P(z, f ), Q(z, f ) be two difference-differential polynomials of f . If f nP(z, f ) = Q(z, f ) holds and if the total degree of Q(z, f ) in f and its derivatives and their shifts is ≤ n, then m(r, P(z, f )) = S(r, f ). Lemma . ([]) Suppose that m, n are positive integers satisfying m + n < . Then there exist no transcendental entire solutions f and g satisfying the equation af n + bgm = , with a, b being small functions of f and g, respectively. Lemma . ([]) Assume that c ∈ C is a nonzero constant, α is a non-constant meromorphic function. Then the differential equation f  + (cf (n)) = α has no transcendental (...truncated)


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Nana Liu, Weiran Lü, Tiantian Shen, Chungchun Yang. Entire solutions of certain type of difference equations, Journal of Inequalities and Applications, 2014, pp. 63, 2014,